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Connexion en topologie fine et balayage des mesures

Bent Fuglede (1971)

Annales de l'institut Fourier

On montre d’abord que la topologie fine est connexe et localement connexe, dans le cas d’un espace harmonique Ω satisfaisant au groupe d’axiomes ( A 1 ) de Brelot (y compris l’axiome de domination). Un autre résultat principal (qu’on n’établit complètement ici que pour le cas classique d’un espace de Green) affirme que, pour toute mesure positive μ sur Ω , soit à support compact, et pour toute base B Ω telle que μ ( B ) = 0 , la mesure balayée μ B a pour support fin la frontière fine de la réunion de toutes les composantes...

Finely differentiable monogenic functions

Roman Lávička (2006)

Archivum Mathematicum

Since 1970’s B. Fuglede and others have been studying finely holomorhic functions, i.e., ‘holomorphic’ functions defined on the so-called fine domains which are not necessarily open in the usual sense. This note is a survey of finely monogenic functions which were introduced in (Lávička, R., A generalisation of monogenic functions to fine domains, preprint.) like a higher dimensional analogue of finely holomorphic functions.

Logarithmic capacity is not subadditive – a fine topology approach

Pavel Pyrih (1992)

Commentationes Mathematicae Universitatis Carolinae

In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be founḋn [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.

Minimal thinness for subordinate Brownian motion in half-space

Panki Kim, Renming Song, Zoran Vondraček (2012)

Annales de l’institut Fourier

We study minimal thinness in the half-space H : = { x = ( x ˜ , x d ) : x ˜ d - 1 , x d > 0 } for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of H below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.

Normal spaces and the Lusin-Menchoff property

Pavel Pyrih (1997)

Mathematica Bohemica

We study the relation between the Lusin-Menchoff property and the F σ -“semiseparation” property of a fine topology in normal spaces. Three examples of normal topological spaces having the F σ -“semiseparation” property without the Lusin-Menchoff property are given. A positive result is obtained in the countable compact space.

On the integral representation of finely superharmonic functions

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri (2019)

Commentationes Mathematicae Universitatis Carolinae

In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset U of a Brelot 𝒫 -harmonic space Ω with countable base of open subsets and satisfying the axiom D . When Ω satisfies the hypothesis of uniqueness, we define the Martin boundary of U and the Martin kernel K and we obtain the integral representation of invariant functions by using the kernel K . As an application of the integral representation we extend to the cone 𝒮 ( 𝒰 ) of nonnegative...

On the sense preserving mappings in the Helm topology in the plane

Pyrih, Pavel (1999)

Serdica Mathematical Journal

∗Research supported by the grant No. GAUK 186/96 of Charles University.We introduce the helm topology in the plane. We show that (assuming the helm local injectivity and the Euclidean continuity) each mapping which is oriented at all points of a helm domain U is oriented at U.

Plurifine potential theory

Jan Wiegerinck (2012)

Annales Polonici Mathematici

We give an overview of the recent developments in plurifine pluripotential theory, i.e. the theory of plurifinely plurisubharmonic functions.

The image of a finely holomorphic map is pluripolar

Armen Edigarian, Said El Marzguioui, Jan Wiegerinck (2010)

Annales Polonici Mathematici

We prove that the image of a finely holomorphic map on a fine domain in ℂ is a pluripolar subset of ℂⁿ. We also discuss the relationship between pluripolar hulls and finely holomorphic functions.

The Pluripolar Hull and the Fine Topology

Armen Edigarian (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that the projections of the pluripolar hull of the graph of an analytic function in a subdomain of the complex plane are open in the fine topology.

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